Transcript uniform circular motion

Circular Motion
5.2 Uniform Circular motion
5.3 Dynamic of Uniform
Circular Motion
HW4: Chapt.5: Pb.23, Pb.24, Pb.30, Pb.33, Pb.36, Pb.53- Due
FRIDAY, OCT. 2
Applications of Newton’s Laws Involving
Friction
Example 5-3: Pulling against friction.
A 10.0-kg box is pulled along a horizontal
surface by a force of 40.0 N applied at a
30.0° angle above horizontal. The coefficient
of kinetic friction is 0.30. Calculate the
acceleration.
Uniform Circular Motion—Kinematics
Uniform circular motion: motion in a circle of
constant radius at constant speed
Instantaneous velocity is always tangent to
the circle.
Linear vs. Circular motion:
Linear motion
Relationship
Angular motion
Position
Angle
Displacement
Angular
displacement
Average velocity
Average angular
velocity
Instantaneous
velocity
Instantaneous
angular velocity
Uniform circular motion
Uniform circular motion means
Constant rotation speed
• How far?
– Angular displacement
Dq = qf - qi
– The size of the angle swept out by
the motion
– Typically “+” indicates counterclockwise
– Units - radians
Go around once:
2π radians
360 degrees
1 revolution

r
y
f
i
x
Uniform circular motion
How fast?
– Angular velocity

y
– This is constant for Uniform
circular motion
– Units: rad/sec
rpm-revolutions per minute
rad/s
r
f
i
x
Relationship between
angular and linear motion
• How far does it go?
– Angular displacement,  to linear
motion, s.
s = rDq
– Here r is the radius of the circle in
meters, and s is the distance traveled
in meters (or arc length).
 is the angular displacement in radians
s
r
= Dq
since s/r is unitless, radians are not
a physical unit, and do not need to
balance like most units.
s
r
y

f
i
x
Relationship between
angular and linear motion
How fast does it go?
– Angular velocity , to linear velocity, v
v avg =
Ds
Dt
=
rDq
Dt
= rw avg
– Direction of v is tangent to the circle
– Units :
v
v m/sec
 must be in rad/s
y
r
v
f
i
x
• A) Objects 1 and 2 have the same linear velocity, v,
and the same angular velocity, .
• B) Objects 1 and 2 have the same linear velocity, v,
and the different angular velocities, .
• C) Objects 1 and 2 have different linear velocities,
v, and the same angular velocity, .
• D) Objects 1 and 2 have different linear velocities,
v, and the different angular velocities, .
Question
• Two objects are sitting on a horizontal
table that is undergoing uniform circular
motion. Assuming the objects don’t slip,
which of the following statements is true?
1
2
Question
• Two objects are sitting on a horizontal
table that is undergoing uniform circular
motion. Assuming the objects don’t slip,
which of the following statements is true?
• A) Objects 1 and 2 have the same linear
velocity, v.
• B) Object 1 has a faster linear velocity
than object 2.
• C) Object 1 has a slower linear velocity
than object 2.
1
2
Period and frequency
w = 2pf
1
T=
f
Linear vs. Circular motion:
Linear motion
x
Position
Displacement
Average velocity
Instantaneous
velocity
Angular motion
Dx = x f - xi
Dx
Dt
Dx
= lim
Dt®0 Dt
vavg =
vinst
Angle
Angular displacement
Average angular
velocity
Instantaneous angular
velocity
q
Dq = q f - qi
Dq
wavg =
Dt
Dq
winst = lim
Dt®0 Dt
Dynamics of Uniform Circular
Motion
Velocity can be constant in magnitude, and we still
have acceleration because the direction changes.
• Direction: towards the center of the circle
Newton’s second law
• Whenever we have circular motion, we
have acceleration directed towards the
center of the motion.
• Whenever we have circular motion,
there must be a force towards the
center of the circle causing the
circular motion.
åF
v
=
ma
2
r
r
ar = rw =
2
r
Dynamics of Uniform Circular Motion
There is no centrifugal force pointing outward;
what happens is that the natural tendency of
the object to move in a straight line must be
overcome.
If the centripetal force vanishes, the object
flies off at a tangent to the circle.
Dynamics of Uniform Circular Motion
Example 5-11: Force on revolving ball
(horizontal).
Estimate the force a person must exert on a
string attached to a 0.150-kg ball to make the
ball revolve in a horizontal circle of radius
0.600 m. The ball makes 2.00 revolutions per
second. Ignore the string’s mass.